IACS Institutional Repository

Indian J. Phys. 82 (10), 1399-1400 (2008) 
d^RTHCOMING PUBLICATIONS 
Volume 82 (2008) 
Interaction of an electric charge in the radial field of a heavy dyon 
GAURAV BHAKUNI AND SHUCHI BISHT 
Class transitions of compact objects Cygnus X-1 and Cygnus X-3 
ACHINTYA K CHATTERJEE, MO WASHIMUL BARI AND ASIT K CHAUOHURY 
Microwave reflection on Echelon cum Lamellar grating and its applications Hi 
astrophysics 
P SENGUPTA, S MONOAL AND J N CHAKRAVORTY 
UV-visible-near IR and infrared spectroscopy of ^carotene and /^carotene-iodine 
complex 
A T OZA, G K SOLANKI, ANAND AMIN AND PARIMAL TRIVEDI 
A new band system of the MgBr molecule 
K N UTTAM, RENU SINGH AND PAVITRA TANDON 
Calculation of the masses and the running masses of the quarks and 
leptons from electroweak to supersymmetric grand unification 
B B DEO AND P K MISHRA 
Self consistent microscopic theory of frictional ratfchet in a nonequilibrium 
environment 
JYOTIPRATIM RAY CHAUDHURI AND DESASHIS BAMK 
Magnetocaioric effect in amorphous ribbon based on finemet 
M M UDO*N, S MANJURA HOQUE, MO SULTAN MAHMUD, 
M A HAKIM AND F U Z CHOWDHURY 
Growth and characterization of gel grown potassium hydrogen levo-tartrate 
crystals 
JAYESH GIVANI MHO MIHIR J JOSHI 
1400 Forthcoming Publication 
Growth, structural and optical studies on organic single crystal imidazole 
P M ANBARASAN, G MEENAKSHI, K JEYAPRIYA, 
M K SUBRAMANIAN AND K SUBRAMAN! 
Evaluation of kinetic parameters from thermoluminescence glow curve 
MAHUA KARMAKAR, BIKASH KUMAR SARKAR, SUPRIYA BARMAN, 
P S MAZUMDAR, S DORENDRAJIT SINGH, W SAMBHUNATH SINGH AND 
MANABESH BHATTACHARYA 
Magnetospheric electron temperatures inferred from whistler observations at low latitudes 
KRISHNA K SINGH, A K SINGH, M ALTAF, M M AHMAD, 
B L KOUL, LALMANI AND R P SINGH 
Validity of theoretical models in estimating the sound velocity in toluene-alcohol 
mixtures 
PALANIAPPAN LAKSHMANAN 
Luminescence studies on 7 - irradiated Dy activated KNaS04 phosphors 
A K UPADHYAY, S J DHOBLE AND R S KHER 


Forthcoming Publications

Indian J. Phys. 82 (11), 1425-1427 (2008) 
Vol.82 • No. II • November 2008 
cpc ONTENTS 
Front Title 1419 
Advisory Board and Editors 1421-1422 
From Editor's Desk 1423 
Front Contents 1425-1427 
Astrophysics, Atmospheric & Space Physics 
Class transitions of compact objects Cygnus X-1 and Cygnus X-3 
ACHINTYA K CHATTERJEE, MD. WASHIMUL BARI AND ASIT K CHAUDHURY 1429-1440 
Microwave reflection on echelon cum lamellar grating and its 
application in astrophysics 
P SENGUPTA, S MONDAL AND J N CHAKRAVORTY 1441-1446 
Magnetospheric electron temperatures inferred from whistler 
observations at low latitudes 
KRISHNA K SINGH, A K SINGH, M ALTAF, M M AHMAD, B L KOUL, 
LALMANI, R P SINGH, JAIPAL SINGH AND BALRAJ KUMAR 1447-1456 
Condensed Matter & Materials Physics 
Magnetocaloric effect in amorphous ribbon based on Finemet 
M M UDDIN, S MANJURA HOQUE, MD. SULTAN MAHMUD, 
MA HAKIM AND F U Z CHOWDHURY 1457-1465 
[ Cont'd, on next page \ 
1426 Contents 
Luminescence studies on y-irradiated Dy activated KNaS04 phosphors 
A K UPADHYAY, S J DHOBLE AND R S KHER 1467-1472 
Growth, structural and optical studies on organic single crystal 
imidazole 
P M ANBARASAN, G MEENAKSHI, K JEYAPRIYA, M K SUBRAMANIAN 
AND K SUBRAMANI 1473-1483 
Growth and characterization of gel grown potassium hydrogen levo-
tartrate crystals 
JAYESH GOVANI AND MIHIR J JOSHI 1485-1494 
Evaluation of kinetic parameters from thermoluminescence glow curves 
MAHUA KARMAKAR, BIKASH KUMAR SARKAR, SUPRIYA BARMAN, 
P S MAZUMDAR, S DORENDRAJIT SINGH, W SAMBHUNATH SINGH 
AND MANABESH BHATTACHARYA 1495-1502 
General & Interdisciplinary Physics 
Validity of theoretical models in estimating the sound velocity in 
toluene-alcohol mixtures 
PALANIAPPAN LAKSHMANAN 1503-1512 
Optics & Spectroscopy 
UV-visible-near IR and infrared spectroscopy of p-carotene and 
p-carotene-iodine complex 
A T OZA, G K SOLANKI, ANAND AMIN AND PARIMAL TRIVEDI 1513-1522 
A new band system of the MgBr molecule 
K N UTTAM, RENU SINGH AND PAVITRA TANDON 1523-1527 
Particle Physics 
Calculation of the masses and the running masses of the quarks 
and leptons from electroweak to supersymmetric grand unification 
B B DEO AND P K MISHRA 1529-1561 
Interaction of an electric charge in the radial field of a heavy dyon 
GAURAV BHAKUNI AND SHUCHI BISHT 1563-1575 
Contents 1427 
Statistical Physics 
Self consistent microscopic theory of frictional ratchet in a 
nonequilibrium environment 
JYOTIPRATIM RAY CHAUDHURI AND DEBASHIS BARIK 1577-1590 
Forthcoming Publications 1591 
Notes for Contributors 1593-1596 
Our Special Publications 1597-1598 
Renewal Notice-2009 1599 
Renewal/New Order Form-2009 1601 
Subscription Rates-2009 1603 
Contents from Back Cover (Cont'd) 1606 


Front Contents

Front Title

Indian J. Phys. 82(1), 131-132 (2008) 
r Special Publications 
•Commemorative Volumes to mark the celebration of 75th year 
of publication of Indian Journal of Physics Volumes 1-6 
1. Frontiers in Materials Physics 
Guest Editor • D Chakravorty 
Price . Rs, 600.00 (10 articles, 247 pages, illustrated) 
Date of Publication 2002 
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Guest Editor Sandip K Chakrabarti 
Price Rs 600 00 (12 articles, 303 pages, illustrated) 
Date of Publication 2002 
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Date of Publication 2003 
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Guest Editors • Amitava Raychaudhurf and Partha Mitra 
Price * Rs 600 00 (11 articles, 281 pages, illustrated) 
Date of Publication • 2004 
5. Frontiers in Condensed Matter Physics 
Guest Editors J K Bhattacharjee and B K Chakrabarti 
Price: Rs. 600.00 (7 articles, 308 Pages, illustrated) 
Date of Publication : 2005 
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Date of Publication : 2005 
*For Volumes 1-6, please contact Publishers: 
Messrs. Allied Publishers Pvt Ltd. 
A-104, Mayapuri, Phase-II, New Delhi-110 064, India 
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751, Anna Salai, Chennai-600 002, India 
132 Our Special Publications 
11. Landmark Papers in Indian Journal of Physics 
ri926-2001J 
Guest Editors : J K Bhattacharjee and 
DSRay 
Price : Rs. 500.00; US $. 50.00 (289 pages, 
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Date of Publication : December, 2002 
III. 75th Year of Discovery of Raman Effect : 
A Special Issue of India Journal of Physics 
[1928-2003] 
Guest Editor: S Chakravorti 
Price : Rs. 500.00; US $. 50.00 (164 pages, 
illustrated) 
Date of Publication : February, 2003 
IV. Production of Radioactive Ion Beams 
[PRORIB-2001] 
Proceedings of the DAE-BRNS sponsored 
International Workshop-A Special Issue of 
Indian Journal of Physics 
Guest Editor : Alok Chakrabarti 
Price : Rs. 200.00; US $. 50.00 (226 pages, 
illustrated) 
Date of Publication : May, 2002 
V Commemoration Volume on Professor K S 
Krishnan: 
Birth Centenary [1898-1998] 
Guest Editors : S Ray and C K Majumdar 
Price : Rs. 100.00; US $. 40.00 (332 pages, 
illustrated) 
Date of Publication : December, 1999 
VIII. XI DAE Symposium ofHigh Energy Physics 
: A Special 
Issue of Indian Journal of Physics 
Guest Editor: Ranabir Dutt and Asim K Ray 
Price : Rs. 300.00; US $. 100.00 (395 pages, 
illustrated) 
Date of Publication : February, 1997 
IX. Lectures of Theoretical Physics Seminar 
Circuit [ 1990-92] : A Special Issue of Indian 
Journal of Physics 
Guest Editors : C K Majumdar and Partha 
Ghosh 
Price : Rs. 80.00; US $. 35.00 (100 pages, 
illustrated) 
Date of Publication : November, 1993 
X. Commemoration Volume on Professor 
Meghnad Saha: Birth Centenary [ 1893-19931 
Editors : H Banerjee and S P Sen Gupta 
Price : Rs. 140.00; US $. 40.00 (240 pages, 
illustrated) 
Date of Publication : December, 1993 
XI. High Temperature Superconductivity : 
Lectures of 
Theoretical Physics Seminar Circuit [1989-
90] : A Special Issue of Indian Journal of 
Physics 
Editors : B K Chaudhuri and S P Sen Gupta 
Price : Rs. 80.00; US $. 30.00 (220 pages, 
illustrated) 
Date of Publication : March, 1992 
XII. Commemoration Volume on Professor C V 
Raman: 
Birth Centenary [1888-1988] 
Editor: S P Sen Gupta 
Price : Rs. 50.00; US $. 25.00 (225 pages, 
illustrated) 
Date of Publication : July, 1989 
VI. Physics and Technology of Accelerators : A 
Special Issue of Indian Journal of Physics on 
the occasion of Golden Jubilee of India's 
Independence 
Guest Editor : Debashis Basu 
Price : Rs. 100.00; US $. 40.00 (236 pages, 
illustrated) 
Date of Publication : December, 1999 
VII. Recent Trends in Statistical Physics : A 
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Indian Journal of Physics 
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Price : Rs. 100.00; US $. 40.00 (KG pages, 
illustrated) 
Date of Publication : August, 1998 
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English

Our Special Publications

Dutta, N

Mohanta, D

Choudhury, A

Inlaw I Plm 82 (3), 327-332 (2008) 
Synthesis and characterizations of quality PbS quantum 
dots in SBR matrix 
N Dutta*, D Mohanta and A Choudhury 
Department of Physics, Tezpur University PC) Napaam 
Sonitpur Assam-784 028 India 
E-mail nabanitaf" iezu ernet in 
Abstract We have successfully grown PbS quantum dots in styrene butadiene rubber (SBR) matrix 
following solution chemistry The formation of monodisperse isolated PbS quantum dots with size distribution 5-
M) nm was confirmed by transmission electron micrographs Structural and optical properties have been 
haractenzed by X-ray diffractometer and photoluminescence spectroscopy Since there is significant 
enhancement \n the band gap of quantum dots by (7-8 times 2 15 eV) compared to bulk PbS (0 29 eV at 300 K), 
PbS quantum dots can be promising candidates for application in optoelectronic sensors/devices operated over 
broid spectrum of light 
Keywords PbS quantum dot, SBR latex, Photoluminescence XRD and TEM 
PACS Nos 81 07 Ta 68 37 Lp, 78 55 -m 61 10 Nz 
1 Introduction 
Over the years, research in low dimensional semiconductor quantum structures, namely 
quantum dots have received significant attention due to size dependent electronic properties 
[1] Worthwhile progress has been achieved in the areas of synthesis, and characterizations 
e g investigating electrical and optical properties of quantum dots [2] These fundamental 
properties of nanoscaled semiconductor structures are strongly influenced by their size 
and dimensionality [3], Such dependence popularly known as 'quantum size effect' offers 
the possibility to manipulate their properties by size and shape control Controlled variation 
of nanocrytallites size results in remarkable changes in the optoelectronic properties from 
molecular to the bulk [4, 5]. Quantum confinement effects are particularly important if the 
crystallite dimension is less than a critical size known as the exciton Bohr radius of the 
material [6]. 
I j Pi 
Corresponding Author 
© 2 0 0 8 I ACS 
328 N Dutta, D Mohanta and A Choudhury 
PbS is an attractive IV—VI binary semiconductor compound having direct and narrow 
energy gap 0.29 eV at 300 K. It has a large bulk Bohr exciton radius (aB ~ 18 nm), small 
carrier effective masses (m* = 0 1 mc) and large optical dielectric constant (r =^172) 
Accordingly, strong quantum confinement effects can be easily noticed and the ground 
state absorption edge would be tunable over a wide range of wavelengths (from near infra 
red to visible). This is because the PbS quantum dots exhibit discrete electronic states 
distributed over a wide energy gap with respect to their narrow band gap bulk counter part 
[7,8]. 
Quantum dots embedded in polymeric matrix provide an interesting way of studying 
nanomatenals. Polymers are considered as good host materials capable of protecting 
nanoparticles from agglomeration [9, 10]. They normally offer long term stability and durability 
against environmental degradation. In this paper, we report the synthesis of PbS quantum 
dots in a flexible and nontoxic matrix namely, styrene butadiene rubber (SBR). Prepared 
quantum dots are characterized by X-ray diffraction (XRD), and transmission electron 
microscopy (TEM). In addition, effort was made to obtain distinct excitonic peaks through 
photoluminescence spectroscopy (PL). 
2. Experimental details 
The carboxylated SBR latex (99% pure, Synthomer, Germany) was selected as matrix to 
encapsulate the PbS uantum dots. SBR is a synthetic rubber co-polymer of butadiene 
and styrene (Figure 1). It is made by the freeradical polymerization in an emulsion systerr 
using redox initiators such as hydrogen peroxide and ferrous sulfate We have decided to 
specifically work in this medium as it provides a flexible polymer system which exhibits 
cistrans geometric isomers and hence capable of accommodating nanoparticles with better 
size dispersity. 
CH 2 — CH — CH 2 — CH = = CH — CH2-
Figure 1. Structural representation of SBR copolymer 
One coat of SBR latex was drawn over glass substrate and left for drying up to 24 
hrs. Next, as received SBR thick films were washed in double distilled water and dipped 
into 0.1 M PbCI2 solution up to 1 hour. Finally, the films containing Pb++ were treated with 
H2S (prepared by Na2S soln in an acidic medium). Thus, PbS quantum dots are grown in 
flexible SBR latex matrix. The size and distribution of the quantum dots were determined 
by TEM'. XRD was performed to obtain the information regarding its structure, crystallinity 
and approximate size (Model : Seifert 3003TT). PL studies has been performed as it 
Synthesis and characterization of quality PbS quantum dots in SBR matrix 329 
provides information relating to different energy states available between valence band 
and conduction band responsible for radiative recombination and indicates luminescence 
efficiency of PbS quantum dots (Amnico Brown fluorescence spectrometer). 
3. Results and discussion 
The TEM micrographs, shown in Figure 2(a) indicate formation of Pbs quantum dots with 
average size 5-10 nm. They are spherical in size and isolated from each other. The 
reasonable amount of inhomogeneity seen in the size distribution could be due to 
nonuniform incorporation of Pb++ into SBR coatings. The advantage is unclusteredness 
thus providing a system to study individual quantum dots, whcih otherwise would be 
iifficult in nanostructured films where grain boundary effect is a serious problem [11] 
25 
20 1 
15 
ion 
5 
0 + r ^ r ^ - r 
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 
Size 
Figure 2(a). TEM micrograph of PbS quantum 
dots 
Figure 2(b). Size distribution vs no of PbS 
quantum dots 
Referring to Figure 3 shows a typical X-ray diffraction Pattern of PbS quantum dots 
embedded in flexible SBR matrix. The XRD pattern contains Bragg's diffraction peaks 
corresponding to reflection from (111), (200) and (220) planes of PbS cubic rock salt 
structure [12, 13]. The diffraction peak signal corresponds to weak intensity owing to low 
concentration of PbS. The average diameter has been estimated from the line width of the 
i — i — i — i — i — i — i — r — T — r 
20 22 24 26 28 30 32 34 36 38 40 44 46 48 
Angle (29) 
Figure 3. XRD spectra of PbS quantum dots 
330 NDutta, D Mohanta and A Choudhury 
diffraction peaks using Scherrer formula (D = 0.9 A//Jcos<p), where A is wavelength of 
X-ray, p is full width at half maxima and <P is the Bragg angle) found to be 9.16 nm 
[14] 
The PL study of the PbS sample reveals three emission peaks corresponding to 573 
nm and 617 nm and 825 nm respectively. The peak at 573 nm is attributed to the band 
edge emission of the PbS uantum dots The second peak is the excitonic peak observable 
at 617 nm Normally, for other binary semiconductor systems (e.g CdS) excitonic emission 
is rarely observable as it is very close to the band edge emission with energy spacing of 
the order of a few meV only 
D 
si
on
 
(A
 
SILL 
LU 
0 09-. 
0 08 \ 
0 07 J 
0 06 J 
0 05 J 
0 04 J 
0 03 J 
0 02 J 
0 01 J 
0 400 800 1200 
Wavelength (rum) 
Figure 4. PL spectra of PbS quantum dots 
On the other hand, in PbS system, upto - 10 fold energy gap enhancement might 
provide an opportunity to notice visible excitonic emission In our case, band edge emission 
( A= 573 nm, E = 2.171 eV) and exxciton emission ( A= 617 nm, E = 2.016 eV) are 
well separated by an energy spacing 155 meV, which is the binding energy of our PbS 
quantum dots. The result of the calculation of excitonic energy in consistency with earlier 
reports [15]. Here the exciton energy is 2eV. Although effective mass approximation (EMA) 
is not valid reasonably in case of PbS [16], however since Eb is less than 100 meV, then 
the combination of small effective mass and large dielectric constant makes the EMA 
useful in case of bulk PbS [17] so using the formulae Eb = ffym*/e2 , the exciton binding 
energy (E^) for PbS bulk system has been calculated [where Ry is the Rydberg energy 
m* is effective mass of electron and e is the high frequency dielectric constant] to be 
0 014 eV. Inour PbS quantum dot system, Eb (0.155 eV) has been increased by over 10 
fid due to significantly large energy gap more than 10 times due to the huge energy gap 
enhancement (from 0.29 eV to 2.15 eV) while undergoing transition from bulk to molecular 
system. The third peak was visible at 825 nm, due to surface state led recombination 
emission. Envelope function calculationof the electronic structure of PbS uantum dots 
also provides the energy spectrum as well as gives selection rules for different transitions 
in uantum dots of smaller diameters [18]. Such calculations indicate that better emissive 
Synthesis and characterization of quality PbS quantum dots in SBR matrix 331 
property of PbS quantum dots is assured by the number of excitons staying at higher 
energy states However, the highest energy state of the excitons is the band edge But 
tor better emissive properties one requires maximum numbers of excitons staying at higher 
energy states [19]. Combination of strong quantum confinement and high dielectric constant 
of PbS quantum dot system enables observation of such distinct excitonic emission peaks 
related to higher energy states [20]. Thus, the PL spectra obtained by us is successful in 
isolating such emission peaks from the band edge emission 
4 Conclusion 
Quality PbS quantum dots have been prepared using a very simple and less expensive 
process The quantum dots have been properly characterized by various techniques The 
possibility of obtaining stable excitons is indicated by PL spectra It may be explored 
further as to its origin and tunability over a given spectrum for applications in diffeent 
optoelectronic devices. 
Increase in available number of excitons in Pbs quantum dot system brings the promise 
of population inversion leading to the realization of quantum dot laser Further, PbS uantum 
dots can be explored for generating emission extended to IR region, which would find 
potential in making detctors operated at IR and telecommunication wavelength 
Acknowledgment 
The authors would like to extend sincere thanks to RSIC, Shillong for TEM, IIT Guwahati 
for PL and XRD facility 
References 
[1] K K Nanda, S N Sarangi, S Mohanty and S N Sahu Thin Solid Films 21 322 (1998) 
[2] Y Wang and N Herron J Phys Chem 95 525 (1991) 
[3] C B Murray, D J Noms and M G Bawendi J Am Chem Soc 11 8706 (1993) 
[4] Frank W Wise Ace Chem Res 33 773 (2000) 
[5} T Tnndade, P O Bnen and N L Pickett Chem Mater 13 3843 (2001) 
[6J A D Yoffe Advances in Physics 42 173 (1993) 
[7] N de la Rosa-Fox, R Erce-Montil la, M Pinero and L Esquivias Optical Materials 22 1 (2003) 
[8] Y Wang, A Suna, W Mahler and R Kawoski J Chem Phys 87 7315 (1987) 
[9] H Du, G Q Xu, W S Chin, L Huang and W Ji Chem Mater 14 4473 (2002) 
[10] D Mohanta, A Choudhury and G A Stanciu A GIT Laboratory J Europe 3 20 (2004) 
[11] I A Ovid'ko Science 295 2386 (1964) 
[12] Shihe Yang, Suhua Wang and K K Fung Pure Appl Chem 72 1223 (2004) 
[13] K Suresh Babu, C Vijayanb and R Devanathan Matenals Letters 58 1223 (2004) 
[14] H P Klug and E Alexander X-ray Diffraction Procedures (New York Wiley) (1974) 
332 A/ Dutta, D Mohanta and A Choudhury 
[15] Huaquiang Cao Guozhi Wang, Sichun Zhang and Xinrong Zhang Nanotechnology 17 (13) 328^ 
(2006) 
[16] R S Kane, R E Cohen and R Silbey J. Phys. Chem. 100 (1996) 
[17] R E Hummel and P Weissmen Hand book of optical properties of small particles and interfaces 
Vol II (New York . CRC Press) (1997) 
[18] K Kang and K Daneshvar J. Appl. Phys. 95 4747 (2004) 
[19] I Kang and F W Wise J. Opt. Soc. Am. B14 1632 (1997) 
[20] Arnim Henglein Chem Rev. 89 1861 (1989) 


Synthesis and Characterizations of Quality PbS Quantum Dots in SBR Matrix

indwl Phys. 82(4),377-378 (2008) 
INDIAN 
JOIINAL of 
PHYSICS 
Vol. 82 • No. 4 • April 2008 
CONTENTS 
Front Title 
Advisory Board and Editors 
From Editor's Desk 
Front Contents 
Astrophysics, Atmospheric & Space Physics 
Estimation of magnetospheric electron densities from low latitude whistlers 
KRISHNA KUMAR SINGH, SHUBHA SINGH, R P PATEL, JAIPAL SINGH, 
A K SINGH, ASHOK K SINGH, R P SINGH AND LALMANI 379-386 
Atomic & Molecular Physics 
Computation of doubly-excited 1 s23/3//1P° and 1F° states in beryllium-like ions 
UTPAL ROY AND Y K Ho 387-401 
General & Interdisciplinary Physics 
A novel high speed fully differential CMOS amplifier 
B CHAKRABORTY AND R R PAL 403-414 
Nonlinear Dynamics & Complex Systems 
Radiation and dissipation effect on unsteady MHD micropolar flow past an infinite 
vertical plate in porous medium with time dependent suction 
MOHAMED M ABDELKHALEK 415-434 
^ra 
373-374 
375 
377-378 
[ Cont'd, on next page ] 
378 Con/enis 
Injection/suction effect on an oscillatory hydromagnetic flow in a rotating horizontal 
porous channel 
K D SINGH AND ALPHONSA MATHEW 435^45 
Plasma Physics 
Large amplitude solitary waves in four component dusty plasma 
PRASANTA CHATTERJEE AND SANJIB KUMAR KUNDU 447-457 
Statistical Physics 
Mixing dynamics and microscopic correlation of associated liquids 
G G BORTHAKUR AND B BAISHYA 459-465 
Erratum 
A minimum electrophilicity perspective of the HSAB principle 
P K CHATTARAJ AND S GIRI 467-468 
Forthcoming Publications 469-473 
Notes for Contributors 475-478 
Our Special Publications 479480 


Indian 7. Phys. 82 (4), 479-480 (2008) 
r Special Publications 
""Commemorative Volumes to mark the celebration of 75th year 
of publication of Indian Journal of Physics : Volumes 1-6 
1. Frontiers in Materials Physics 
Guest Editor : D Chakravorty 
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480 Our Special Publications 
II Landmark Papers in Indian Journal of Physics 
[1926-2001] 
Guest Editois J K Bhattacharjee and 
D S Ray 
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Date of Publication December, 2002 
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A Special Issue of India Journal of Physics 
11928-2003] 
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Date of Publication February, 2003 
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1PRORIB-2001J 
Proceedings of the DAE-BRNS sponsored 
International Workshop-A Special Issue of 
Indian Journal of Physics 
Guest Editoi Alok Chakrabarti 
Price Rs 200 00, US $ 50 00 (226 pages, 
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Knshnan 
Birth Centenary [1898-1998] 
Guest Editors S Ray and C K Majumdar 
Price Rs 100 00, US $ 40 00 (332 pages, 
illustrated) 
Date of Publication December, 1999 
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Special Issue of Indian Journal of Physics on 
the occasion of Golden Jubilee of India's 
Independence 
Guest Editor Debashis Basu 
Price Rs 100 00, US $ 40 00 (236 pages, 
illustrated) 
Date of Publication December, 1999 
VII Recent Trends in Statistical Physics A 
Special Issue of 
Indian Journal of Physics 
Guest Editor J K Bhattacharjee 
Price . Rs 100 00, US $ 40.00 (140 pages, 
illustrated) 
Date of Publication • August, 1998 
VIII XI DAE Symposium of High Energy Physic s 
A Special 
Issue of Indian Journal of Physics 
Guest Editor Ranabir Dutt and Asim K Rj\ 
Price Rs 300 00, US $ 100 00(395 pages 
illustrated) 
Date of Publication February, 1997 
IX Lectures of Theoretical Physics Seminai 
Circuit [1990-92] A Special Issue of Indian 
Journal of Physics 
Guest Editors C K Majumdar and Partha 
Ghosh 
Puce Rs 80 00, US $ 35 00 (100 pages 
illustrated) 
Date of Publication November, 1993 
X Commemoration Volume on Professor 
Meghnad Saha Birth Centenary [ 1893-1993 | 
Editors H Banerjee and S P Sen Gupta 
Price Rs 140 00, US % 40 (X) (240 pages 
illustrated) 
Date of Publication December, 1993 
XI High Temperature Superconductivity 
Lectuiesof 
Theoretical Physics Seminar Circuit |1989 
90| A Special Issue of Indian Journal of 
Physics 
Editors B K Chaudhun and S P Sen Gupta 
Price Rs 80 00, US % 30 00 (220 pages 
illustrated) 
Date ot Publication March, 1992 
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illustrated) 
Date of Publication July, 1989 
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Devi, O Babynanda

Singh, C Amuba

Indian J. Phys. 82 (5), 551 -560 (2008) 
Dirac and Infeld-Hull ladder operator methods for a 
modified oscillator potential 
O Babynanda Devi* and C Amuba Singh 
Department of Physics, Manipur University, 
Canchipur, Imphal-795 003, Manipur, India 
E-mail amuchab(citediffmail com 
Abstract Dirac's method of solving the Schrodinger eigenvalue problem for the harmonic oscillator by 
constructing eigenvalue raising and lowering operators for a potential of the form V(x) = Ax*' + Bx -1 is reviewed 
The eigenvalue problem of the potential is also solved by applying the Infeld and Hull's factorization method In the 
process we construct the Infeld- Hull ladder operators which connect eigenvectors of a hierarchy of Hamiltonians 
The laddering actions in the Dirac method and Infeld - Hull method are compared and constrasted 
Keywords : Dirac ladder operators, Infeld-Hull ladder operators, Infeld-Hull factorization 
PACS Nos. . 03.65.-w, 03.65.Fd 
1. Introduction 
In the early days of quantum mechanics Dirac[1] developed the elegant operator method 
of finding the energy eigenvalues and eigenvectors of the harmonic oscillator by constructing 
the eigenvalue raising and lowering operators. This celebrated method could be found in 
most text books [2] on nonrelativistic quantum mechanics. Later, generalizing Schrodinger's 
approach of 'solving eigenvalue problems by factorization' [3], Infeld and Hull [4] developed 
an operator factorization method which enables us to solve the eigenvalue problem of a 
large class of second order differential equations algebraically. In this method the original 
differential operator is a function of a parameter m and the factorization essentially provides 
a laddering process which takes a solution for certain value of m to solutions for m ± 1. 
Even as the factorization method of Infeld and Hull is very versatile and, in fact, lies at 
the very foundation for the latter day development of the Supersymmetric Quantum 
Mechanics [5], as for the linear harmonic oscillator system the Dirac approach is more 
Corresponding Author 
©2008IACS 
552 O Babynanda Devi and C Amuba Singh 
straightforward and directly generates the higher eigenfunctions from the ground-state 
eigenfunction. Moreover in the Dirac method the normalization constants are easily 
determined without having to evaluate cumbersome integrals. 
In [6] we have extended Dirac's method to find the energy eigenvalues and normalized 
eigenfunctions for the potential V(x) = Ax2 + Bx 2. Section 2 gives a brief review of this 
work.. In Section 3 we apply the Infeld- Hull factorization method to the above potential 
In Section 4 we factorise the Hamiltonian in four different ways thereby identifying four 
different Infeld-Hull ladder operators and their laddering actions is explained. A detailed 
comparison of the laddering actions in the Infeld - Hull method and the Dirac method is 
given. The comparison also explains why the Dirac ladder operator for the system is 
necessarily quadratic in the momentum. 
The present work also carries an Appendix A where we discuss the revelation of 
unphysical spectrum when the requirement of normalisability is relaxed. 
2. Dirac ladder operator method for the potential V(x) = Ax2 + Bx 2 
For a Hamiltonian H Dirac ladder operators 0± are linear operators, which satisfy the 
commutation relations 
[H, 0 ± ]= ±cQ± (1) 
where e is a real constant having dimension of energy and signifies the quantum of 
energy by which the ladder operator would shift the energy eigenvalue. The basic laddering 
property of Q± is that - if W E = EVE, then H(Q± ¥E) = (E±e)(Qt V f ) , provided Qi 
does not annihilate VE . Obviously, Q+ and 0_ are essentially the hermitian conjugates of 
each other. Since £ is a parameter independent of the state of the physical system 
existence of Dirac ladder operators implies that the Hamiltonian has equi-spaced eigenvalue 
spectrum. And it is obvious that Dirac ladder operators may not exist for an arbitrary 
Hamiltonian. Our definition, eq. (1), also rules out state-dependent ladder operators which 
are constructed adhoc, such as the ones in Refs. [7, 8]. As in [6] we consider a harmonic 
oscillator subjected to a repulsive force varying as A-3, which is described by the 
Hamiltonian 
H = ~ ^ ( p 2 + x2 + ^ ()3-1)x-2) f (2) 
where x is the dimensionless position variable and p = -id/dx and P is a dimensionless 
parameter (P > 1). The eigenvalue problem can be algebraically solved [6 ] by constructing 
the ladder operators O and Of, where 
0 = 9 1 H/ho)-ixp-)C - - , 
2J 
(3) 
Dirac and Infeld-Hull ladder operator methods for a modified oscillator potential 
These operators H, Q and Q* satisfy the commutation relations 
[H, Or ] = 2 ho) Q\ [H, O] = -2h<D Q [Q Of ] - 4H/»w . 
553 
(4) 
That is, Cfl and O are Dirac ladder operators respectively raising and lowering the 
eigenvalues of H by 2hco. Factorizing the Hamiltonian as 
( '\\ 1 
H = hco\ A^A+0+ I,
 Wjth >4= (p-/x+/j3/x) 2 I . / o - - ' " ' (5) 
and using the algebra eq.(4) it can be shown [6] that the energy eigenvalues are 
En=Pio)\ 2n + p + 1 ^ 
and that the normalized eigenfunctions are 
(6) 
Vn(x) = CnxPe'**C~2 n = 0, 1, 2, 3, .. (7) 
where l?n{z) is the Laguerre polynomial [9] and Cn =• 2n\/r P + 1 ^ 
The reduced radial Schrodinger equation for a central potential V(r) -Ar2 + Br2 is 
identical, in form, to the one dimensional Schrodinger eigenvalue equation for the potential 
V(x) =A x2 + Bx'2 . Therefore the energy eigenvalues and normalized wavefunctions 
obtained above can be readily adapted for this central potential without having to solve the 
Schrodinger equation. An added advantage of this method over the method of direct solution 
[10,11] of the Schrodinger equation is that the normalization constants of the wavefunctions 
are automatically determined once we normalize the groundstate wavefunction. 
3. Application of the Infeld-Hull factorisation 
Treating /} as a parameter we rewrite the Hamiltonian of eq. (2) in the factorised form as 
with 
We then have 
H(fi) = ho>(AHP)A(P) + p + ^ 
A(P) = J-(p-ix + ipx'). 
«P)AHP)tM±V_fi+l 
h(x) 2 
(8) 
(9) 
(10) 
554 O Babynanda Devi and C Amuba Singh 
The eigenvalue equation H(P)lFE = EX¥E can be written as 
AHP)A(P)VE=[*-P-1 *E> 01) 
where, A = E/ftco. (12) 
Applying A(P) on both sides of eq. (11 ) and using eq. (10 ) we obtain 
H08 + 1) A(P)V'E = {E-hio) A(P)VE . (13) 
That is, A(p) acting on the eigenvector YE of H(fi) produces an eigenvector of 
H(/J + 1) belonging to the eigenvalue (E-hco). Further eq. (11) gives 
{*pyE\*p)rE) = [x-p-iyEp>E). (14) 
Assuming y £ to be normalizable A(ft) VE is also normalizable provided A > 0 + 1/2 
that is, if E> (ft + 1/2) hto. Introducing the vectors 
*/».«Hi = MP + m) ®ium< m = 0, 1, 2, 3, 
with 4>p = f s , it is easy to show that 
H(ft + m)<Pp,m =(E-m h<o)<t>p,m . 
Then eq. (14) is rewritten as 
(^+i|^+i)=[A-^-^](^|fE)-
Similarly, 
The process can be formally continued to get 
(15) 
(16) 
(17) 
( * * m l K m * ) « ( A - l 8 - 5 - ^ 08) 
Dirac and Infeld-Hull ladder operator methods for a modified oscillator potential 555 
To avoid the possibility of the norm-square becoming negative the process of successive 
generation of the 0 - vectors has to terminate at some stage, say at m = n and that can 
happen only if 
A = £ + - + 2n, n = 0, 1, 2, 3 
that is, if E = / t o f0 +
 2 + 2n j ; (19) 
and in that case A{(3 + n) 0p+n = 0. 
Eq. (19) gives the eigenvalues of the Hamiltonian H{P). Obviously the ground-state 
Hf0 of H(P) is defined by the requirement A(P)H'0 = 0. 
4. Dirac ladder operatos vs Infeld-Hull ladder operators 
We have seen that in the Dirac approach (Section 2) the ladder operator O1 takes an 
eigenstate of H(P) to the next higher eigenstate of the same Hamiltonian. In the 
factorization method of Infeld and Hull the operator A(P) takes an eigenstate of H(p) to 
an eigenstate of H(P +1). 
The Hamiltonian 
H(P)^ho)^(p2^x2^P(P-^x~2) 
can be factorized in four different ways (including the one given in eq. (8)) : 
H(P) = tnofA*(P)A(P) + fi+l\
 ( 2 0 a ) 
H(P) = hto B\P)B{P)-P-±Y (20b) 
H(P) = hw(cHP)C(P) + P-^) (20c) 
and, H(P) = hco\DHP)D(P)-P+^ (20d) 
where, A(P) = -j=(p-ix + ip x"1), B(/3) = -^ (p+/ i f+ / i8 x~1), (21) 
556 O Babynanda Devi and C Amuba Singh 
and C = -^=(p+/x- / ( /3-1)x- 1 ) , and D = - ^ ( p - / x - / ( 0 - 1 ) x - 1 ) . (22) 
If V £ is an eigenvector of H{(1) belonging to the eigenvalue E we have seen that 
(vide eq. (13)) 
H(P
 + 1) A{/3)YE =(E-fia>) A(P)YE , 
Similarly it can be easily shown that 
H(P +1) B(P)>FE = (E + hw) B(P) YE , 
H(P-^C(PWE = (E + h(o)C{P)TE, 
W - 1 ) D ( / l ) ! P E = ( E - A f i ) ) ^ ) ! P E l 
A/»+D 
v..,W-l *.(/» + !> 
*.(/»-!)' 
%(/»-!) 
W + D 
H{p-\) H(P) H(fi+l) 
%{0 + 2) 
H{p + 2) 
(23a, 
(23b) 
(23c) 
(23d) 
Figure 1. Laddering actions of Dirac ladder operators Q and Qf vs laddering actions of the Infeid-Huli 
ladder operators A, B, C and D. 
Dirac and Infeld-Hull ladder operator methods for a modified oscillator potential 557 
That is, A(p), B(/3), C((3) and D(/3) are Infeld-Hull ladder operators which take an 
eigenvector of H(/3) to an eigenvector of H(fi + 1) or H(p -1) with eigenvalues increased 
or decreased by hco . These Infeld-Hull ladder operators are linear in the momentum p. 
The actions of these four Infeld-Hull ladder operators and the Dirac ladder operators Q 
and Of are illustrated in Figure 1. It will be seen that while the Dirac ladder operators are 
Vertical1 (enabling to go up and down the eigenvectors of H(P)) the Infeld-Hull ladders are 
'inclined* ones taking an eigenvector of H(P) to an eigenvector of H(P±1). Figure 1 
also illustrates that the Dirac ladders Q and O t can also be constructed as 
O t=C(i8+1)f l ( iB) = / l t( i9)S(/5) l (24a) 
0=D( /?
 + 1 ) ^ ) = S t ( ) 8 ) ^ ) , (24b) 
and also as 
O t =e( /5 -1 )C( i8 ) = D t(i8)C(j8) f (24c) 
0=>KjS-1)D(/3) = C t(/3)D(j8). (24d) 
5. Conclusion 
In Section 2 we have given the Dirac ladder operators which are quadratic in momentum 
operator can be constructed only for a potential of the form V(x) = Ax2 + Bx2 and we 
determine the energy eigenvalue and normalize eigenfunctions . In Section 3 we have 
worked out the Infeld and Hull's factorization method to determine the energy eigenvalues 
for this particular system of modified oscillator with a view to compare it with the Dirac 
laddering method. The comparison is done in Section 4 to point out the contrasting features 
as well as the relationship underlying the two types of ladder. The laddering process in 
the Dirac approach is among the eigenvectors of the particular Hamiltonian whereas in 
the Infeld - Hull method the laddering process connects the eigenvectors of a hierarchy of 
Hamiltonians H(/J-1), H(/3), H(/5 + 1) 
As we conclude we observe that for the eigenvalue problem for the modified oscillator 
we have considered here the Dirac approach is simpler and more direct than the 
factorization method of Infeld and Hull. However the Dirac approach, though elegant, lacks 
versatility, - besides the harmonic oscillator potential and the modified oscillator potential 
we have considered here, we do not know of any other potential which would directly 
admit the Dirac ladder operators. 
Acknowledgments 
Oinam Babynanda Devi gratefully acknowledges Council of Scientific and Industrial 
Research, New Delhi, Government of India, for grant of research fellowships (No: 09/ 
476(0032)/2004-EMR-l). 
558 O Babynanda Devi and C Amuba Singh 
References 
[1] P A M Dirac The Principles of Quantum Mechanics (4th ed) (Oxford Oxford University Press) o 
VI p 136 (1958) 
[2] (a) J L Powell and B Crasemann Quantum Mechanics (USA Addison-Wesley, Inc), Ch 9 Sec 9 
14, p 321 (1965) 
(b) R L Liboff Introductory Quantum Mechanics (USA Addison-Wesley, Inc), p183 (1980) 
[3] E Schrodinger Proc Roy Irish Acad 46A 183 (1941) 
[4] L Infeld and T E Hull Rev Mod Phys 23(1) 21 (1951) 
[5] Fred Cooper, Avinash Khare and Uday Sukhatme Supersymmetry in Quantum Mechanics (Singapore 
World Scientific) (2001) 
[6] C Amuba Singh and O Babynanda Devi Indian J Phy 79(2) 131 (2005) 
[7] Shi-Hai Dong, R Lemus and A Frank Int J Quantum Chem 86 433 (2002) 
[8] Shi-Hai Dong and R Lemus Int J Quantum Chem 86 265 (2002) 
[9] I S Gradshtein and I M Ryzhik Table of Integrals, Senes and Products (London Academic Press 
p1037 (1960) 
[10] I I Gol'dman, V D Knvchenkov, V I Kogan and V M Galitskii Problems in Quantum Mechanic 
(London Infosearch Ltd) p8 63 (1960) 
[11] L D Landau and E M Lifshitz Quantum Mechanics (Oxford Pergamon Press) ChV, Sec 36 pl28 
(1977) 
[12] S Brajamani and C A Singh J Phys A Math Gen 23 3421 (1990) 
Appendix A: Existence of unphysical spectra 
As stated in Section 4 the Hamiltonian 
H(0) = ftw-(p2 + x2+ 0(0-1) x~2) 
can be factonzed in four different ways 
H(P) = ha)(AU + p + ±Y (A1) 
HtfJ-AcoftfB-jS-l]. 
H(j3) = / ia ; (c t C+/ 3 - -
(A.2) 
(A.3) 
and H(P)=*a>\DfD-P + l \ (A.4) 
Dirac and In feld-Hull ladder operator methods for a modified oscillator potential 559 
where, the operators A S, C, and D are defined in eqs. (21) and (22) in the text. The first 
of these factorizations has been employed in the Dirac approach of Section 2 as well as 
in the Infeld-Hull approach of Section 3 and it leads to the physical spectrum 
En = ho) (2n + P +1/2) (with our choice of p > 1). 
In both the Dirac approach and the Infeld-Hull method the crucial input is the 
normalizability of the eigenvectors and the preservation of the normalizability under the 
action of the various operators introduced. If we leave aside this condition of normalizability 
just for the sake of consideration, the above four factorizations of H(P) reveal the existence 
of spectra enumerated in Table 1. 
Table 1. Physical and unphysical spectra 
Factorisation Eqn for V'0(x) lP0(x) Y'n(x) Eigenvalue Normalisability* 
Eqn (A 1) Af'0(x) = 0 x°e *** (Q t ) 'V 0 (x ) M 2n* /J + - Normalizable 
if >1 
V ( 1 \ 
Eqn (A.2) BFQ(x)^0 x V ( O ) > 0 ( * ) - M 2 n + /J + - Not 
normalizable 
- t~. ,. 1 
Eqn (A.3) GPo(x) = 0 X^e2 (O)n^0(x) - t o 2 n + / N g ] Not 
normahsable 
Eqn(A,4) OfQ(x) = 0 x 1 ~V*** ((fft'oix) r,(,,[ 2 n + ' h 2 Normahsable 
if < 0 
"Normalizability is the condition for physical realizability 
For a positive value of K = P(p~V we may choose either (1 > 1 or p < 0. 
)8> 1 means P = A s l ( l W l + 4K) , and 
)3< 0 means 0 = & s V i - V l + 4K) . 
In the text throughout we have chosen /? > 1 and therefore obtained the physical 
spectrum in the top row in Table 1, while the remaining three spectra cannot be physically 
realized. Had we chosen P < 0 the physically realizable spectrum would be the one 
given in the bottom row of Table 1, while the other three would remain unrealizable. Of 
course these two physical spectra (in Table 1, top row for p = ft > 1 and the bottom row 
for p = ft, < o) and their eigenfunctions are exactly identical because ft + ft = 1. Similarly 
560 OBabynanda Devi and CAmuba Singh 
the unphysical spectra in the second row for 0 = ft is also identical with the unphysicai 
spectra in the third row for 0 = ft, • The existence of these unphysical spectra with no 
lower bound can be traced to the fact that the operators r, = 1/4(0+Q1> 
T2 = il4\Q-CP] and r 3 = G/2hu) form a second order realization[12] of the su (11; 
Lie algebra. 


Dirac and Infeld-Hull Ladder Operator Methods for a Modified Oscillator Potential

Choudhury, Joydeep

Sarkar, Nirmal Kumar

Bhattacharjee, Ramendu

Indian J. Phys. 82 (5), 561-565 (2008) 
IP 
Algebraic approach to analyze the vibrational spectra of 
tetrahedral molecules 
Joydeep Choudhury1 , Nirmal Kumar Sarkar2 and Ramendu Bhattacharjee1* 
1
 Department of Physics, Assam University, Silchar-788 Oil, Assam, India 
1
 Deparment of Physics, Kanmganj College , Kanmganj-788 710, Assam, India 
E-mail choudhuryjoyC" rediffmail com 
Abstract Molecular spectroscopy is undergoing a considerable attention in last few decades There is a 
considerable current research interest in the study of vibrational ground and excited states of polyatomic 
molecules using Lie algebraic method Both the stretch and bend fundamental vibrational modes of tetrahedral 
molecules are calculated by using U(2) algebraic model Its application to four molecules are presented The 
result obtained by this model are the best fit with the observed data This study will give a general approach for 
solving tetrahedral molecules with good accuracy 
Keywords Algebraic model, vibrational energy level, vibrational mode, tetrahedral molecules 
PACS Nos. 03 65 Fd , 31 15 Hz 
1. Introduction 
Molecular spectroscopy has received a lot of attention in last few decades. There is 
considerable current research interest in the study of vibrational ground and excited states 
of polyatomic molecules using Lie algebraic method. U(4) and U(2) algebraic model have 
been used so far in the analysis of experimental data. The algebraic approach to molecules 
was first introduced by Wulfman [1] in the study of Morse oscillator, lachello, Levine and 
co-workers [2,3] described the rotation-vibration spectra of molecules using U(4) algebra. 
Later on the U(4) algebra was used for each bond in case of linear triatomic and four 
atomic molecules [5,6]. It is to be noted that the U(4) model becomes complicated when 
the number of atoms in a molecule increases more than four. In 1984, Van Roosmalen 
et a/, proposed an algebraic model to describe stretching vibrational modes of ABA 
molecules [7]. This model is based on the isomorphism between one dimensional Morse 
potential and SU(2) algebra and correspond to the algebraic version of coupled Morse 
Corresponding Author 
©2008IACS 
562 Joy deep Choudhury, Nirmal Kumar Sarkar and Ramendu Bhattacharjee 
oscillator method developed by Halonen and Child [8]. Michelot, Moret -Bailly and Leroy 
[9,10] proposed an algebraic approach to study the vibrational stretching mode m a 
polyatomic molecule using unitary group U(n) with (n -1) vibrational degrees of freedom 
In order to study vibrational modes of medium sized molecules, one usually chooses a 
tetrahedral molecules as one of good samples. In 1994, Lemus and Frank proposed an 
algebraic SU(2) model based on group theoretical ideas to calculate the stretching and 
bending vibrational energy levels of methane like molecules[12]. Recently, using Lie 
algebraic method we have reported better results [11,22] for the vibrational frequencies of 
HCN than those reported earlier. The U(2) model was used successfully in explaining the 
stretching vibrations of polyatomic molecules like octahedral and benzene molecules [13,14] 
.Later on, lachello and Oss has developed the algebraic model to incorporate the bending 
modes in case of benzene [15] and acetylene [16]. As the algebraic approach emphasizes 
anharmonicity (and cross anharmonicity) even in the zeroth order, higher order coupling 
can be introduced in a systematic and sequential fashion. As such the approach is 
particularly appropriate for many challenges of modern spectroscopy. This approach is 
important for the representation of higher accuracy spectroscopic data and especially so 
far larger molecules. The algebraic approach starts with a Hamiltonian and hence such a 
fit provides more than just a compact parametrization of the data. Rather, it determines 
some of the parameters in the Hamiltonian and so provides explicit predictions (including 
information on the potential). 
In this paper, the fundamental vibrational modes of tetrahedral molecules like GeH4 
GeCI4 ,SiF4 and CCI4 are calculated by using an algebraic model Hamiltonian. The values 
of boson numbers N1 and N2 can be found out by knowing anharmonic constant ioe and 
coexe of stretching and bending bonds of different diatomic molecules under study. The 
result obtained by this model are in good agreement with the observed data. Therefore 
this study will give "a general approach" to solve the vibrational energy levels accurately 
for tetrahedral molecules. 
2. Algebraic model 
The model is based on the isomorphism of the U(2) Lie algebra and one dimensional 
Morse oscillator whose eigen states may be associated with U(2) D 0(2) states[13]. For 
Figure 1. Assignment of the Ui (2) algebra to tetrahedral molecule. 
Algebraic approach to analyze the vibrational spectra of tetrahedral molecules 563 
a tetrahedral molecule, AB4 , we introduce four U,(2) (1 < / < 4) algebra to describe A-
B interactions and six 11,(2) algebra (5 < j < 10) for B-B interaction [Figure 1] 
The two possible chains of molecular dynamical groups in tetrahedral molecules are 
given by the product 
U1(2) x .. .x U10(2) 3 01(2) x ... . xO10(2) D 0(2) (21) 
U1(2)x x U10(2) 3 U(2) 3 0(2). (2.2) 
Which corresponds to local and normal coupling respectively. For these two situations 
the Hamiltonian operator can be diagonahsed analytically 
The Hamiltonian upto two body interactions is given in terms of 10 operators, 
representing ten one dimensional Morse oscillator, plus two types of bond-bond interaction 
C2(0,j(2)) and M|j(2) known as Casimir and Majorana operators respectively of OIJ(2) 
group. 
For tetrahedral molecules , the Hamiltonian can be written as 
H = Hs+ HB+ Vs-*. (2 3) 
The term Hs denotes the stretching degrees of freedom (A-B bond), HB is the bending 
contribution (B-B bond) and Vs - B represents the stretching bending interaction which is 
neglected in case of first order approximation The common algebraic model Hamiltonian 
in case of stretching and bending mode for tetrahedral molecules can be considered as 
[13,17] 
n n n 
"=E0 + £4C,+£4yC,y + XAA (24) 
( - 1 I(I >() 
Where / runs from 1 to n = 4 for calculation of stretching and / runs from 5 to n = 10 for 
bending vibrational energy level. In the eq. (2 4) , C, is an invariant operator with eigen 
values 4{v? - A/, v,) and the operator C is diagonal with matrix elements 
(A/, l l / /;A/y ,^|C ( / |A/„V /;A/y ,^) = 4[ ( i / / + ^ ) 2 - ( ^ + ^)(A/ / + /Vy) (2.5) 
while the operator Mu has both diagonal and non-diogonal matrix element. 
(A/,, v- Nt, v, \M„\ N„ v,\ Nr Vj, = (N,v, + N,v, -2v,v,) 
(A/„ v, +XNr v, -1 |M„ N„ v,;Nr v, =-[v,(v, +1)(/VI -vl)(Nj -v, +1)]V 2 (26) 
(N„ v, - 1 ; Nr v, +1|M,| N„ v,\ Nr v,) = -[v,(vt +1) (N, - v,)(A/, - v, +1)]V 2 
564 Joydeep Choudhury, Nirmal Kumar Sarkar and Ramendu Bhattacharjee 
For a tetrahedral molecule AB4, we choose the parameters A/,= Nv At= A, (for / -
1 to 4) , Av= Ay2 (any / and j ) and A 12 for stretching mode and A/y = A/2, At= A5 (for , 
= 5 to 10) , An = A56 (any / and y) and A5 6 for bending mode respectively. 
3. Results and discussion 
We report in Table I the results of fundamental vibrational energies Av F2 (stretching) and 
E ,F2 (bending) for four tetrahedral molecules like GeH4, GeCI4 ,SiF4 and CCI4 taken 
from references 18,19,20 and 21. The estimated values of six parameters (A,, A12, A 
for stretching and A5 , A5 6 , A 5 6 for bending mode) along with boson numbers N1 and 
N2 are also shown in the Table 1 for four molecules . We have reported the vibrational 
fundamental modes v
 v v2> v 3 and v 4 for four tetrahedral molecules , but we can 
obviously calculate overtone and combination bands up to higher number of quanta as 
well. The model Hamiltonian (2.4) appears to describe the vibrational frequencies with 
good accuracy. 
Table 1. Comparison of calculated and observed fundamental vibrational frequencies of GeH4 GeCi 
,S(F4 and CCI4 
Normal 
mode of 
vibration 
GeH4 GeCL S»R CCI4 
1 ^1 
1 v2 
1 V3 
1 v4 
A i 
E 
F2 
F
2 
Stretching Parameters 
2106 2105 88 
930.9 930.94 
2111.78 2111.66 
819 3 819 31 
GeH4 
398 
132 
450 
172 
398 04 
132.35 
449 94 
172 33 
GeCI4 
800 8 
264.2 
1031.39 
388 44 
SiF 
800 58 
264.20 
1031.3 
388 45 
4 
460 
2142 
792 
3135 
CCI 
459 96 
214 5 
791 83 
312 9 
4 
N, 
A, 
A12 
^ 
Bending Parameters 
N* 
A 5 
A 5 . 6 
*S.6 
55 
-8.215 
0.2534 
0 0263 
28 
-29 475 
2 5595 
-0.9968 
299 
-0 34 
-0.0010 
0 0434 
140 
-1 
-0.0958 
0.0714 
100 
-1 175 
0.1404 
0 5769 
15 
-13.94 
1.113 
2.070 
342 
-0 155 
-0 0431 
0 2426 
140 
-1 
-.0765 
0.1758 
All values are in cm1, except Nt and N2 which are dimensionless. Observed values 
are taken from references 18, 19, 20 and 21. 
Algebraic approach to analyze the vibrational spectra of tetrahedral molecules 565 
4. Conclusions 
Using model Hamiltonian, we have calculated the vibrational fundamental modes of four 
tetrahedral molecules, GeH4: v1 = 2105.88, v2 = 930 94, \'3 =r 2111 66, r4 = 819 31 cm 
1
 GeCI4 : V) = 398.04, v2 = 132.35, v3 =449 94, r4 = 172 33 cm1 , SiF4 v, = 
800 58, v>2 = 264.2, v3 =1031 3 , n4 = 388 45 cm"1 and CCI4 ^ = 459 96, v2 s 214 5, 
v3 =791.83, v4 = 312.9 cm"1 with good accuracy As it is already reported that AB4 is 
an important member of the family of molecules. Hence, we can claim that using this 
model ,the vibrational fundamental modes of some tetrahedral molecules can be found out 
accurately. It is obvious that once we calculate the vibrational fundamental modes of a 
molecule accurately, the overtone and combination bands can be calculated with the help 
of such a model Hamiltonian by fitting the experimental data with the help of computer 
programming. 
Acknowledgment 
One of us (RB) is thankful to the DST, government of India for providing research grant 
References 
[1] R D Levine and C E Wulfman Chem Phys Lett 60 372 (1979) 
[2] F lachello Chem Phys Lett 78 581 (1981) 
[3] F lachello and R D Levine J Chem Phys 77 3046 (1982) 
[4] F lachello and R D Levine Algebraic Theory of Molecules (Oxford Oxford University press) (1995) 
[5] S Oss Adv Chem Phys 93 455 (1996) 
[6] F lachello, S Oss and R Lemus J Mol Spectroscopy 146 56 (1991) ibid 149 132 (1991) 
[7] O S Van Roosmalen, I Benjamin and R D Levine J Chem Phys 81 5986 (1984) 
[8] L Halonen and M S Child Mol Physics 46 239 (1982) 
[9] F Michelot and J Moret- Bailly J Phys AS 51 (1987) 
[10] C Leroy and F Michelot J Mol Spectroscopy 151 71 (1992) 
[11] N K Sarkar, J Choudhury and R Bhattacharjee Mol Phys 104 3051 (2006) 
[12] R Lemus and A Frank J Chem PhysWI 8321 (1994) 
[13] F lacnello and S Oss Phys Rev Lett 66 2976 (1991) 
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Synthesis and Characterizations of Quality PbS Quantum Dots in SBR Matrix

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Dirac and Infeld-Hull Ladder Operator Methods for a Modified Oscillator Potential

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